#7230866

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(3,4)(6,7)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,29,33)(27,30,34,38)(28,31,36,32)(35,37,40,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27)(26,30)(28,35)(29,34)(31,37)(32,39)(33,38)(36,40);(1,3,5,4)(2,6,8,7)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(1,4,5,3)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(27,34)(30,38)(35,40)(37,39);(2,8)(6,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7138062);